1. Pale Blue Dot is a photograph of planet Earth taken on February 14, 1990, by the Voyager 1 space probe from a record distance of about 6 billion kilometers, as part of the Family Portrait series of images of the Solar System.

2. Presently, the earliest known archaeological evidence of any form of writing or counting are scratch marks on a bone from 150,000 years ago. But the first really solid evidence of counting, In the form of the number one, is from a mere twenty-thousand years ago. An ishango bone was found in the Congo with two identical markings of sixty scratches each and equally numbered groups on the back. These markings are a certain indication of counting and they mark a defining moment in western civilization.1

3. In Suma, around 4,000 BCE, Sumerians used tokens to represent numbers.
An advantage of using tokens to represent numbers is that in addition to adding tokens you can also take away, giving birth to arithmetic, an event of major significance.
The Sumerian’s tokens is of great importance in our society, because it made it possible to assess wealth, calculate profit and loss and to collect TAXes!!!!.
Keep permanent records.
This is the world’s first writings and accounting system.
In comparison: First step pyramid …....around 2800 BC

4. The Babylonian number system.
Around 1900 BC
The Babylonian number system.
The Babylonians `wrote' on wet clay by pressing a stylus or wedge into it in one of a small number of ways. The Babylonian number system, already from a very early era, was remarkably sophisticated.
It was positional, They had to remember only 2 Symbols instead of ten and had the base ten without the zero.

5. Babylonian number system had a base of 60, inherited from the Sumerian and Akkadian civilizations, and possibly motivated by the large number of divisors that 60 has. The sexagesimal measurement of time and of geometric angles is a legacy of the Babylonian system.

7. Number “1” was written with a single stroke `stroke'
Numbers 2 through 9 were written by combining multiples of a single stroke:
The number 10 was written in a single character
For example, 11 was written

8. 1500 BCE The Babylonians invented trigonometry
1500 BCE The Babylonians invented trigonometry. The cuneiform inscriptions on
Plimpton 322 suggest the Babylonians used a form of trigonometry based on the
ratios of the sides of a triangle rather than the more familiar angles, sines, and cosines.

9. Babylonian tablet YBC 7289 showing the sexagesimal number 1;24,51,10 approximating √2
YBC Yale Babylonian Collection

10. The ten fingers on two hands, is a possible starting point of the decimal counting as observed with children.

11. Decimal notation is a base 10, positional notation
Decimal notation is a base 10, positional notation. It is the Hindu-Arabic numeral system.
It can refer to non-positional systems such as Roman or Chinese numerals which are also based on powers of ten.

14. The world's earliest decimal multiplication table was made from bamboo slips, dating from 305 BC, during the Warring States period in China.

16. Positional system , Decimal,
The base or Radix is 10, The number is presented as:
10n An n-1 An n-2 An A A0
Example:
A number such as 3471 is written as
Normally the coefficients are written and the positions are assumed.
More general:
10n An n-1 An n-2 An A A A A
decimal point

17. Positional Binary numbers,
The base or radix is 2, so there is only two numbers 0 and 1 ↓
2n An n-1 An n-2 An A A0
Example:
A number such as (1310) is written as:
=1310
Normally the coefficients are written and the positions are assumed.
More general:
2n An + 2n-1 An n-2 An A A A A
binary point

18. Common number systems that we might need:
Binary base = 2
Octal base =8
Decimal base =10
Hexadecimal base =16

19. Exabyte (2 60 ), Yottabyte ( 2 70) , Xenottabyte (2 80) ,
Binary Prefixes or symbols
The prefix represents the power of 1000, and means 10n
Kilo or K =1,024 bits or , 103
Mega M =1,048,576 bits or 106
Giga G = 1,073,741,824 bits or 109
Tera T= 1,099,511,627,776 bits or
Petabyte ( bytes)
Exabyte (2 60 ), Yottabyte ( 2 70) , Xenottabyte (2 80) ,
Shilentnobyte (2 90) , Demengemgrottebyte (2 100 )…

20. Weight of the digital information storage
1 bit:  A binary decision
1 byte:  A single character,, (8 bits)
10 bytes:  A single word
1 K Byte: A paragraph
2K Bytes A page
100 K Bytes: A very low resolution Photograph
1 M Byte: A small Novel
2 M byte: A high resolution Photograph
5 M byte: A complete works of Shakespeare
10 M byte: A minute of High fidelity sound
100 M bytes: A meter of Shelfed books
800 M bytes: A CD-ROM
1 Giga Bytes: A pick up truck filled with paper
20 G Bytes: A good collection of works of Beethoven
1 Tera byte : ,000 trees made into printed paper
10 Tera bytes: The printed collection of the US Library of Congress.

21. Binary Addition Example 1: Augend 0 1 0 1 1 + Addend 0 1 0 1 0
Results
==========================================================
Augend = =1110
Addend = =1010
Results = = 2110
Example 2:
Augend
Addend
Augend = =1010
Addend = =1410
Results = = 2410

22. Binary Subtraction Example 1: Minuend 0 1 0 1 1 - Subtrahend 0 1 0 1 0
Results
Minuend = =1110
Subtrahend = =1010
Results = =0110
Example 2:
Minuend
Subtrahend
Minuend = =1110
Subtrahend = =610
Results = = 510

23. Binary Multiplication
Example 1
Multiplicand
Multiplier
Results
===========================================
Multiplicand = 2710
Multiplier = 510
Results =13510
Example 2
Multiplicand
Multiplier
Results
===========================================
Multiplicand =
Multiplier = 710
Results =13310

24. Number conversion: Decimal to Binary
Example: Convert the number To Binary
Integer Quotient Remainder Binary Coefficient
52/ ao=0
26/ a1 =0
13/ a2 =1
6/ a3 =0
3/ a4 =1
1/ a5 = 1
The final number is a5 a4 a3 a2 a1 a0 or
=5210

25. Number conversion: Decimal to Binary
Example: Convert the number To Binary
This is done in two steps
First the number is converted and then the decimal part
Integer Quotient Remainder Binary Coefficient
139/ ao=1
69/ a1 =1
34/ a2 =0
17/ a3 =1
8/ a4 =0
4/ a5 = 0
2/ a6 = 0
1/ a7 = 1
The final number is a7a6a5 a4 a3 a2 a1 a0 or = 13910

26. Decimal to Binary , the decimal part:
Example: to Binary
Integer Fraction
0.7 5 *
0.5 0 *
The result is
So the Total conversion result of Decimal to Binary is : is

27. Number Conversion Decimal to octal
Octal is base 8 numbers expressed as follows:
8n An n-1 An n-2 An A A0
Where A can be 0,1,2,3,4,5,6,7
Example: Convert to Octal:
Integer Fraction
343/
42/ 5/
Results = 5278
The answer is from the Fraction part of the procedure
Checking the results 5 * * * 80
5* * =34310

28. Number Conversion Hexadecimal is base 16 number system as follows:
0,1,2,3,4,5,6,7,8,9,A,B,C,D,F
It is used in communicating Binary numbers, for ease of use.
Where Decimal Binary
A = ,
B = ,
C= ,
D= ,
E= , F=

29. Number Conversion
Example: Convert the Hexadecimal number AD7B16 to Binary
AD7B16 =
Example: Convert the Following Binary number to Hexadecimal
Starting from the right separate each 4 bits as follows, add zero to
the left if necessary:
E B A
Answer is 3E5B8A

30. Table 0f Numbers with different bases Decimal Binary Octal Hexadecimal
0000 1
0001 2
0010 3
0011 4
0100 5
0101 6
0110 7
0111 8
1000 10 9
1001 11
1010 12 A
1011 13 B
1100 14 C
1101 15 D
1110 16 E
1111 17 F
Table 0f Numbers
with different bases

31. Negative binary number representation for n=3, 2 𝑛 = 8 values
Decimal
number
Binary
Sign and Magnitude
One’s Complement
Two’s Complement
000 1
001 2
010 ve ve ve 3
011 4
100 5
101 6
110 ve ve ve 7
111
+0 and –0
Hardware Implementation
problem
0nly one “0”
Simple Hardware
Overflow Detection
Please note that the Most Significant Bit (MSB) for a +ve number is 0 and the +ve number does not change. ONLY the the –ve number has its MSB as 1 and the magnitude changes according to the scheme used.

32. An easier method to get the negation of a number
in 2's complement is as follows:
Sign bit
+ve number
Example 1
Example 2
1. Starting from the right, find the first '1'
2. Invert all of the bits to the left of that one
Advantage of the 2’s complement number system is that we can use the same hardware circuit to do the addition and the subtraction
Sign bit
- ve , 2’s complement number

33. Addition with Binary Numbers
Possibility of digits addition:
0+ 0 = 0
0+1=1+0 = 1
1+1 = carry=1 sum=0
4-bit unsigned number
Example_1:
Add A= Augend
B= Addend
Sum=
Example_2:
Add A= Augend 1310
B= Addend 610
Sum= because the overflow is discarded,
overflow

34. Subtraction with binary
Possibility of digits Subtraction:
0 - 0 = 0
1 - 1 = 0
0 - 1 = Borrow =1 Difference =1 =1
4-bit unsigned number
Example_1:
Minuend A =
Subtrahend B =
Difference
Example_2:
Add A= Minuend
B= Subtrahend
Sum= ? Or 1510
Overflow ? Incorrect results

35. Multiplication with Binary Numbers
Example: Multiply the following two 4-bit unsigned binary numbers
A=1101 =1310
B=1001 = 910
multiplicand
multiplier
partial products =

36. Subtraction with 2’complement
Example_1
Perform the following operation:
A-B, A&B are 5 bit signed numbers
A= B=
1’ Complement
1 plus 1
2’ complement of B B
Add
= +610
Subtraction is performed as addition
Example_2
Perform the following operation:
B-A, A & B are 5 bit signed numbers
A= B=
2’s of A = A’+1 = = 10111
’ complement of A B
result is a -ve number
to get its value, complement and add 1
0101 1
Sign bits

37. Addition and subtraction with overflow condition
With 2’s complement addition and subtraction
Two conditions can prevail. When both numbers are +ve or –ve.
Here, overflow occurs and has to be detected
Example_1 Both numbers are +ve.
Add A + B where A= 01011, B= 01100
Both A & B are 5 bit +ve signed numbers S
Result is –ve, indicating an overflow condition
Example_2 Both numbers are –ve and in 2’s complement form
Add A + B where A= 10011, B= 10010
Both A & B are 5 bit -ve signed numbers
Sign
carry
Result is +ve, indicating an overflow condition
1)--ve numbers must be initially in 2’complement form
2)End around carry is neglected

38. Binary Coded Example: Decimal Code, BCD Decimal BCD 0000 1 0001 2 0010
0000 1
0001 2
0010 3
0011 4
0100 5
0101 6
0110 7
0111 8
1000 9
1001
Decimal
BCD 10 11 22 13 64 75 99
217
738
599

39. Addition with BCD Example _1 Example _2 Example _3 Add 310 +410= 710
0011
0100
= 710
Example _2
Add = 1510
0111
1000
= This is not BCD
0110
= 1510
Example _3
Add = 16510
= 1510
= 16510
Numbers greater than 1001 are converted to BCD by adding 0110 to them or
in decimal numbers greater than 9 we add 6 to them to convert them to BCD

40. Some useful codes 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Decimal Digit
Binary
BCD
Excess-3
Gray code
0000
0011 1
0001
0100 2
0010
0101 3
0110 4
0111 5
1000 6
1001 7
1010 8
1011
1100 9
1101 10
unused
1111 11
1110 12 13 14 15
Some useful codes

41. The Control Characters are for routing data and arrangement of
B7 b6 b5
b4b3b2b1
000
001
010
011
100
101
110
111
NUL
DLE SP @ P ‘ p
SOH
DC1 ! 1 A Q a q
STX
DC2 “ 2 B R b r
ETX
DC3 # 3 C S c s
EOT
DC4 $ 4 D T d t
ENO
NAK % 5 E U e u
ACK
SYN
& 6 F V f v
BEL
ETB ’ 7 G W g w BS
CAN ( 8 H X h x HT EM ) 9 I Y i y LF
SUB * : J Z j z VT
ESC + ; K [ k FF FS ,
< L \ l | CR GS - = M ] m } SO RS .
> N ^ n ~ SI US / ? _ o
DEL
ASCII Characters
American Standard
Code for Information
Interchange
The Control Characters
are for routing data
and arrangement of
the text and the printing
of the characters